In many fields today, analog and digital signals are sampled at spaced time intervals to form digital representations for storage, analysis, processing, transmission, reproduction, and other uses. These signals may include, but are not limited to, sounds, images, time-varying measurement values, sensor data such as bioelectrical data like electro-encephalography data (EEG), electrocardiography data (ECG), electromyography data (EMG), electrolaryngraphy data (ELG), electro-oculography data (EOG), control system signals that control other devices or systems, and telecommunication transmission signals. The signal measurements are often intended to depict the state of an object for measurement, which may be a patient's brain as in an EEG, or a picture of the earth as in a seismic survey. Therefore, in each case, it is desirable to obtain an acceptable-quality reconstruction of the signal. The term “acceptable-quality reconstruction” herein refers to a reconstruction with a level of precision sufficient to depict the state of the object faithfully for the selected application.
Today, many fields have their own accepted practice for the number of measurements, commonly expressed as a “sampling rate”, required to obtain an acceptable-quality reconstruction. Many of them are restricted by the Nyquist-Shannon theorem, which states that in order to reconstruct a signal without aliasing artifacts, the signal must be sampled at above twice the bandwidth or the highest frequency component of the signal in the case of baseband sampling, which sampling rate is commonly referred to as the “Nyquist rate.” To reduce the sampling rate below the current practice or below the Nyquist rate while still obtaining an acceptable-quality reconstruction would have many benefits such as reduced costs and measurement time.
Throughout computational science and engineering, several methods are known to represent data in such a parsimonious way. To reduce the sampling rate below the Nyquist rate, mathematical models are proposed in which the major features of the data are represented using an expression with only a few terms, in other words, models that use a “sparse” combination of generating elements taken from a set. A representation is called t-sparse if it is a combination of only t elements. Such mathematical models make use of the sparse nature of the data of interest. The terms sparse (and “sparsity”) here mean any data that have a small number of dominating terms in the model or representation compared to the number of all the terms. These mathematical models consist of a sparse combination of terms taken from a plurality of functions in the expression. When the plurality of functions forms a basis, the sparse combination becomes unique. By allowing a larger set of functions, different sparse combinations may represent the data, e.g. a plurality of different combinations may provide equivalent representations of the data. Often, modelling problems deal with a mixture of diverse phenomena, and therefore a plurality of sparse combinations of terms from different sources or bases may be useful. In addition, nonlinear models are possible that consist of a quotient of sparse linear combinations.
Besides the accuracy of a representation, a model's sparsity has really become a priority. The degree of sparsity affects the achievable level of compression, whether in sampling or in reconstruction. A sparser model means a higher degree of compression of the data, less collection of the data, as well as reduced storage needs or transmission of the data and a reduced complexity of the mathematical model for analysing such data. It may be assumed hereinafter that the given data behave more or less in accordance with a sparse combination of elements taken from a specific set. The aim is then to determine both the support of the sparse combination and the scalar coefficients in the representation, from a small or minimal amount of data samples. It makes no sense to collect vast amounts of data merely to compress these afterwards. Ideally the required data samples may not depend on the specific object that one is dealing with and contain enough information to reconstruct it. Sparse techniques may therefore solve the problem statement from a number of samples proportional to the number of terms in the representation rather than the number of available data points or available generating elements for the model.
Prior art methods utilizing the sparse characteristic to reduce the measurement rates include compressed sensing, see for example U.S. Pat. No. 7,646,924, and finite rate of innovation, see for example U.S. Patent Application No. US 2010/0246729. Each approach has its advantages and limitations. One method may be more suitable in one application while another method is more suitable in another application. Thus, it would be highly desirable to have multiple methods that can give an acceptable-quality reconstruction of the signal from a reduced number of measurements compared to the current practice.
In compressed sensing, down-sampling is performed randomly, hence introducing a probabilistic element which may cause the reconstruction to fail. The gain in samples offered by the technique also comes at a price: the complexity of the optimization algorithms used to recover an approximation to the original signal is higher than the traditional FFT-based algorithms using Nyquist-rate based sampling. In the finite rate of innovation technique the sampling does not take place in the time or spatial domain. Also, the sampling is aimed at picking up the pulses in the signal and so noise significantly influences the result.
In coding theory however, the reconstruction of a t-sparse object in a higher dimensional space may theoretically be achieved using only 2t samples, which is the absolute minimum. With one more sample it is even possible to reveal the correct value of t. But it is widely believed, that a similar result does not hold in a noisy numeric environment, among other things because the decoding algorithm finds the support of the sparse representation by rooting a polynomial, which may be an extremely ill-conditioned problem.
While the present disclosure uses signal processing as the example for illustrating the invention, one skilled in the art will understand that the invention is not limited to the field of signal processing but to the collection, processing, and reconstruction of all data that demonstrate sparsity.